We investigate preconditioned conjugate gradient methods for kernel ridge regression (KRR) problems with a moderate to large number of data points ( 10^4 ≤ N ≤ 10^7 ). We develop and analyze two randomized preconditioners with complementary guarantees. For full-data KRR, RPCholesky preconditioning requires 𝒪(N^2) arithmetic operations for fixed accuracy under sufficiently rapid eigenvalue decay of the kernel matrix. For restricted KRR with k≪ N centers, KRILL preconditioning requires 𝒪((N+k^2)klog k) operations with no eigenvalue-decay assumption. Experiments on benchmark and scientific data sets demonstrate the robustness of both methods relative to existing preconditioners.
We propose fully discrete numerical methods, high-order in both space and time, for solving parabolic equations on semi-unbounded domains. The spatial discretization uses a spectral method based on Laguerre functions, while the temporal discretization employs two distinct hp-version Galerkin time-stepping schemes: the continuous Petrov-Galerkin and the discontinuous Galerkin methods. A priori error estimates are derived in the L^2(0,T;H^1(ℝ_+)) -norm, with all constants explicitly independent of the temporal and spatial discretization parameters. In the presence of singularities at t = 0 , exponential rates of convergence in time are obtained by using geometrically refined time meshes together with approximation degrees that increase linearly. Numerical experiments are conducted to verify the theoretical findings.
This paper examines an energy, momentum, and angular momentum conserving (EMAC) algorithm for the closed-loop geothermal system. We use the finite element method for spatial discretisation and the Crank-Nicolson method for time discretisation, where the first nonlinear term of the geothermal system is processed by the EMAC algorithm. Theoretical analysis shows that the EMAC algorithm yields superior error estimates compared with the commonly used skew-symmetric (SKEW) algorithm: the error of the EMAC algorithm depends on the Reynolds number ( Re ), while the error of the SKEW algorithm depends on Re^3 . Three numerical experiments verify the convergence, effectiveness, and physical conservation properties of the algorithm, and confirm that the L^2 error of the velocity field u obtained by the EMAC algorithm is better than that obtained by the SKEW algorithm as Re increases.
This paper focuses on the optimal error estimates of a fully discrete decoupled scheme for the electrohydrodynamics (EHD) model, which couples the Navier–Stokes equations for velocity and pressure with a transport equation for the charge density and a Poisson equation for the electric potential. By combining the stabilization method and some subtle implicit-explicit treatments for nonlinear coupling terms in time, and mixed finite element methods in space, we develop a linear and decoupled fully discrete scheme. The scheme is proved to be uniquely solvable, charge-conservative, and unconditionally energy stable. By deriving the L^∞ -bound of the numerical charge density, and utilizing the energy argument, we establish the optimal error estimates for all variables rigorously. Various numerical experiments are provided to demonstrate the theoretical results.
This paper introduces a novel approach for estimating the infinity norm of the inverse of monotone (inverse-positive) matrices and totally non-negative matrices. The proposed method is based on a partition of the index set into two complementary subsets. By considering partial row sums associated with this partition, the problem is reduced to the analysis of an auxiliary 2 × 2 matrix that captures the aggregated interaction between the subsets. The main result shows that if the constructed 2 × 2 matrix preserves the monotonicity property, then an explicit upper bound for the infinity norm of the inverse can be derived. This reduction principle enables greater flexibility in applications, as the partition can be adapted to the structure of the matrix, such as sparsity or block patterns. Furthermore, the new bounds generalize existing row-sum-based estimates, which can be viewed as the limit case corresponding to a trivial partition. The approach is extended to totally non-negative matrices via a standard sign transformation, preserving the structure of the obtained bounds. Numerical examples demonstrate that the proposed estimates can be significantly sharper than classical bounds, highlighting their practical relevance in stability analysis and numerical linear algebra.
